Bounds on Characteristic Polynomials

Abstract

Suppose G is a simple graph with n vertices, m edges, and rank r. Let χGptq " a0t n ´a1t n´1 `¨¨¨`p´1q r art n´r be the chromatic polynomial of G. For q, k P Z and 0 ď k ď q `r `1, we obtain a sharp two-side bound for the partial binomial sum of the coefficient sequence, that is,Indeed, this bound holds for the characteristic polynomial of hyperplane arrangements and matroids, and its weak version can be generalized to the characteristic polynomial of toric arrangements and arithmetic matroids. We also propose a problem on the geometric interpretation of the above bound.

Stanley-Reisner rings for quasi-arithmetic matroids

“…As stated in the introduction, in the case of matroids, algebraic structures associated to matroids have been used to prove inequalities for their f and h-vectors. For arithmetic matroid, only very little is currently known about the shape of these vectors [53]. It would be interesting to prove stronger inequalities.…”